LAMINAR DISSIPATIVE FLOW IN A POROUS CHANNEL BOUNDED BY ISOTHERMAL PARALLEL PLATES

IntroductionTheproblemofforcedconvectioninaporousmediumchannelorductisaclassicalone (atleastforthecaseofslugflow (Darcymodel) .Therehasrecentlybeenrenewedinterestintheproblembecauseoftheuseofhyperporousmediainthecoolingofelectronicequipment.Recently ,NieldandBejan[1]refertomorethan 3 0papersonthetopic ,butnoneofthemdealsexplicitlywiththecaseofthermaldevelopment.ThisgapintheliteraturehasbeenpartlyfilledbyNieldetal.[2 - 4 ].Lahjomrietal.[5 ,6 ]havesolvedmathematicallysimilarproblemsbyusingthe…     

Thermally developing flow in finned double‐pipe heat exchanger

A numerical solution of the convective heat transfer in the thermal entry region of the finned double‐pipe is carried out for the case of hydro‐dynamically fully developed flow when subjected to uniform wall temperature boundary condition. Adaptive axial grid size is used in order to cater for the variation of large solution gradients in the axial direction. It has been observed that the thermal entrance region is highly effective and there is a substantial enhancement in the heat transfer coefficient. A maximum of 76.4877% increase has been observed in the thermal entrance region as compared with the fully developed region for 24 fins and H*=0.6 when R?=0.25, whereas for R?=0.5 the maximum increase is 75.0308% for the same number of fins of same height. It has been observed that no geometry consistently perform better throughout the entrance region. However, the geometries that have optimal performance in the fully developed region perform better in the developing region on average terms. Results show that the Nusselt number and the thermal entrance length are dependent upon various geometrical parameters such as ratio of radii of the inner and the outer pipe, fin height and the number of fins. The limiting case results match well with the literature results. This validates our numerical procedure and computer code. Copyright © 2010 John Wiley & Sons, Ltd.     

Forced convective heat transfer in porous medium of wire screen meshes

The hydrodynamic and heat transfer characteristics of a porous medium consisting of 20 wire screen meshes are examined theoretically and experimentally. The hydrodynamic experiments are conducted for the range of Reynolds number based on mean velocity and wire diameter from 1.5 to 12. The Ergun's constants and thermal dispersion coefficients are calculated in this range. Nusselt number variation is determined in both thermally developing and fully developed flows by the help of forced convection heat transfer experiments conducted for the uniform heat flux boundary condition. Correlation functions of Nusselt number in the range of fully developed and thermally developing, and of thermal entrance length are obtained from experimental data. Solutions of momentum and energy equations simulating the experimental model are obtained numerically with variable porosity and the anticipated thermal dispersion coefficients. The thermal dispersion coefficients well-adjusted to the experimental data are determined by numerical solution of the energy equation. Received on 22 November 1996     

LAMINAR DISSIPATIVE FLOW IN A POROUS CHANNEL BOUNDED BY ISOTHERMAL PARALI,EL PLATES

The effects of viscous dissipation on thermal entrance heat transfer in a parallel plate channel filled with a saturated porous medium, is investigated analytically on the basis of a Darcy model. The case of isothermal boundary is treated. The local and the bulk temperature distribution along with the Nusselt number in the thermal entrance region were found. The fully developed Nusselt number, independent of the Brinkman number, is found
to be 6. It is observed that neglecting the effects of viscous dissipation would lead to the well-known case of internal flows, with Nusselt number equal to 4.93. A finite difference numerical solution is also utilized. It is seen that the results of these two methods, analytical and numerical, are in good agreement.     

Forced convection in tightly coiled ducts: Bifurcation in a high Dean number region

The present numerical study is on the fully developed bifurcation structure of forced convection in a tightly coiled duct of square cross-section and curvature ratio of 0.5 in a high Dean number region. Ten solution branches, two symmetric and eight asymmetric, are found. Among them, one symmetric branch and seven asymmetric branches have not been reported in the literature. On these new branches, the flow has a structural 2-, 4-, 5-, 6-, 7- or 8-cell. The mean friction factor and Nusselt number are different on various solution branches. In tightly coiled ducts, the secondary flow enhances the heat transfer more significantly than the friction increase.     

Convective heat transfer in the thermal entrance region of finned double-pipe

Numerical simulation of the steady and laminar convection in the thermal entry region of the finned annulus is carried out for the case of hydrodynamically fully developed flow when subjected to uniform heat flux thermal boundary condition. Finite difference based marching procedure is used to compute the numerical solution of the energy equation. The results to be presented include Nusselt number, as a function of dimensionless axial length and thermal entrance length for various configurations of the finned double-pipe. The numerical results show that Nusselt number has complex dependence on the geometric variables like ratio of radii, fin height, and number of fins. A comparison of the computed results for certain limiting cases with the results available in the literature validates the numerical procedure used in this work.     

Thermally developing flow in curved square ducts with internal fins

The laminar incompressible hydrodynamically fully developed and thermally developing flow is studied in a curved square duct with four longitudinal fins. The duct is successively subjected to constant wall temperature, to circumferentially uniform temperature and axially linearly or exponentially varying temperature. The local and fully developed Nusselt numbers are examined for various values of the Dean number and it is found that the heat transfer rate increases for high fins. The parameters that affect the entry length are studied and the fluctuations of the local Nu that appear in the entrance region are investigated. Temperature contour plots are presented for the visualization of the temperature field and functional relations for the Nusselt number are proposed in terms of the Dean and Prandtl numbers.     

Laminar flow heat transfer for simultaneously developing velocity and temperature profiles in ducts of rectangular cross section

Summary A numerical method is used to solve the heat transfer equations for laminar flow in ducts of rectangular cross section with simultaneously developing temperature and velocity profiles, both for constant wall temperature and for constant heat input per unit length of the duct. Like the solutions for a fully developed velocity profile, the Nusselt number for each aspect ratio is found to increase from a limiting value at large distances from the entry plane to a maximum at the entry plane. The results also show a strong effect of the Prandtl number on the heat transfer coefficients with uniform and fully developed velocity profiles representing the upper and lower limits respectively. Comparisons are made with analytical solutions for circular ducts and parallel plates and with experimental data.     

Experimental studies on heat transfer and friction factor characteristics of CuO/water nanofluid under turbulent flow in a helically dimpled tube

An experimental investigation on the convective heat transfer and friction factor characteristics in the plain and helically dimpled tube under turbulent flow with constant heat flux is presented in this work using CuO/water nanofluid as working fluid. The effects of the dimples and nanofluid on the Nusselt number and the friction factor are determined in a circular tube with a fully developed turbulent flow for the Reynolds number in the range between 2500 and 6000. The height of the dimple/protrusion was 0.6 mm. The effect of the inclusion of nanoparticles on heat transfer enhancement, thermal conductivity, viscosity, and pressure loss in the turbulent flow region were investigated. The experiments were performed using helically dimpled tube with CuO/water nanofluid having 0.1%, 0.2% and 0.3% volume concentrations of nanoparticles as working fluid. The experimental results reveal that the use of nanofluids in a helically dimpled tube increases the heat transfer rate with negligible increase in friction factor compared to plain tube. The experimental results showed that the Nusselt number with dimpled tube and nanofluids under turbulent flow is about 19%, 27% and 39% (for 0.1%, 0.2% and 0.3% volume concentrations respectively) higher than the Nusselt number obtained with plain tube and water. The experimental results of isothermal pressure drop for turbulent flow showed that the dimpled tube friction factors were about 2-10% higher than the plain tube. The empirical correlations developed for Nusselt number and friction factor in terms of Reynolds number, pitch ratio and volume concentration fits with the experimental data within ±15%.     

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

An analytical skin friction and heat transfer model for compressible, turbulent, internal flows

An analytical skin friction model for compressible, turbulent, internal, fully developed flow involving adiabatic and non-adiabatic, smooth and rough flows has been developed by extending the incompressible law-of-the-wall relation to compressible cases. The formula recovers Prandtl's incompressible law of friction for pipes (within 2%) for incompressible flow. The model also shows good correlation with available data for compressible, adiabatic flows and flows involving cold wall heat transfer (within 15%). Comparison with hot wall data is only moderate (15–30%). Finally, using Reynold's analogy, the Stanton number and Nusselt numbers may be estimated.     

Numerical solutions of pulsating flow and heat transfer characteristics in a pipe

Numerical studies are made of flow and heat transfer characteristics of a pulsating flow in a pipe. Complete time-dependent laminar boundary-layer equations are solved numerically over broad ranges of the parameter spaces, i.e., the frequency parameter β and the amplitude of oscillation A. Recently developed numerical solution procedures for unsteady boundary-layer equations are utilized. The capabilities of the present numerical model are satisfactorily tested by comparing the instantaenous axial velocities with the existing data in various parameters. The time-mean axial velocity profiles are substantially unaffected by the changes in β and A. For high frequencies, the prominent effect of pulsations is felt principally in a thin layer near the solid wall. Skin friction is generally greateer than that of a steady flow. The influence of oscillation on skin friction is appreciable both in terms of magnitude and phase relation. Numerical results for temperature are analyzed to reveal significant heat transfer characteristics. In the downstream fully established region, the Nusselt number either increases or decreases over the steady-flow value, depending on the frequency parameter, although the deviations from the steady values are rather small in magnitude for the parameter ranges computed. The Nusselt number trend is amplified as A increases and when the Prandtl number is low below unity. These heat transfer characteristics are qualitatively consistent with previous theoretical predictions.     

Experimental studies of a double-pipe helical heat exchanger

An experimental study of a double-pipe helical heat exchanger was performed. Two heat exchanger sizes and both parallel flow and counterflow configurations were tested. Flow rates in the inner tube and in the annulus were varied and temperature data recorded. Overall heat transfer coefficients were calculated and heat transfer coefficients in the inner tube and the annulus were determined using Wilson plots. Nusselt numbers were calculated for the inner tube and the annulus. The inner Nusselt number was compared to the literature values. Though the boundary conditions were different, a reasonable comparison was found. The Nusselt number in the annulus was compared to the numerical data. The experimental data fit well with the numerical for the larger heat exchanger. But, there were some differences between the numerical and experimental data for the smaller coil; however these differences may have been due to the nature of the Wilson plots. Overall, for the most part the results confirmed the validation of previous numerical work.     

Laminar thermally developing flow in rectangular channels and parallel plates: uniform heat flux

Numerical simulations were conducted for thermally developing laminar flow in rectangular channels with aspect ratios ranging from 1 to 100, and for parallel plates. The simulations were for laminar, thermally developing flow with H1 boundary conditions: uniform heat flux along the length of the channel and constant temperature around the perimeter. In the limit as the non-dimensional length, x* = x/(D _h RePr), goes to zero, the Nusselt number is dependent on x* to the negative exponent m. As the non-dimensional length goes to infinity the Nusselt number approaches fully developed values that are independent of x*. General correlations for the local and mean heat transfer coefficients are presented that use an asymptotic blending function to transition between these limiting cases. The discrepancy between the correlation and the numerical results is less than 2.5 % for all aspect ratios. The correlations presented are applicable to all aspect ratios and all non-dimensional lengths, and decrease the discrepancy relative to existing correlations.     

A two-phase model for laminar film condensation from steam-air mixtures in vertical parallel-plate channels

Detailed results are presented for laminar film condensation from steam-air mixtures flowing downward in vertical flat-plate channels. The mixture flow is laminar and saturation conditions prevail at the inlet. A fully coupled implicit numerical approach is used that achieves excellent convergence behavior, even for high inlet gas mass fractions. The detailed results include velocity, temperature, and gas mass fraction profiles, as well as axial variations of film thickness, pressure gradient and Nusselt number. The effects of a wide range of changes in the four independent variables (the inlet-to-wall temperature difference and the inlet values of gas concentration, Reynolds number, and pressure) on the film thickness, axial pressure gradient, and the local and average Nusselt numbers are carefully examined. It was found that increases in inlet concentration of noncondensable gas caused significant decreases in the film thickness, local Nusselt number, and axial pressure gradient. An analytical solution for the film thickness and velocity field at the end of condensation path was developed and shown to be the asymptotic value of the numerical results for large distances along the channel.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

Pressure drop and heat transfer characteristics of turbulent flow in annular tubes with internal wave-like longitudinal fins

Measured were pressure drop and heat transfer characteristics with uniform axial heat input using air as the working fluid in both the entrance and fully developed regions of annular tubes with wave-like longitudinal fins. Five series of experiments were performed for turbulent flow and heat transfer in the annular tubes with number of waves equal to 4, 8, 12, 16 and 20, respectively. The test tube has a double-pipe structure with the inner blocked tubes as an insertion. The wave-like fins are in the annulus and span its full width. The friction factor and Nusselt number in the fully developed region were obtained. The friction factor and Nusselt number can be well corrected by a power-law correction in the Reynolds number range tested. In order to evaluate the thermal performance of the longitudinal finned tubes over a plain circular tube, comparisons were made under three conditions: (1) identical pumping power; (2) identical pressure drop and (3) identical mass flow. It was found that under the three constraints all the wave-like finned tubes can enhance heat transfer with the tube with wave number 20 being superior. Finally, discussion on the enhancement mechanism is conducted and a general correlation for the fully developed heat transfer is provided, which can cover all the fifty data of the five tubes with a mean deviation of 9.3%.     

Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.     

Effect of particles on turbulent thermal field of channel flow with different Prandtl numbers

The direct numerical simulation(DNS) of heat transfer in a fully developed non-isothermal particle-laden turbulent channel flow is performed.The focus of this paper is on the modulation of the particles on turbulent thermal statistics in the particle-laden flow with three Prandtl numbers(P r = 0.71,1.5,and 3.0) and a shear Reynolds number(Reτ = 180).Some typical thermal statistics,including normalized mean temperature and their fluctuations,turbulent heat fluxes,Nusselt number and so on,are analyzed.The results show that the particles have less effects on turbulent thermal fields with the increase of Prandtl number.Two reasons can explain this.First,the correlation between fluid thermal field and velocity field decreases as the Prandtl number increases,and the modulation of turbulent velocity field induced by the particles has less influence on the turbulent thermal field.Second,the heat exchange between turbulence and particles decreases for the particle-laden flow with the larger Prandtl number,and the thermal feedback of the particles to turbulence becomes weak.